Young's Inequality in Semifinite von Neumann Algebras

Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $\tau$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(\mathcal{M},\tau)$, with $0<p<\infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(\mathcal{M},\tau)$ can be represented in the form $P(x)=\Phi(x^m)$ $(x\in L^p(\mathcal{M},\tau))$ for some continuous linear m...

In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.

We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and R\'enyi's relative entropy known in finite dimension.

In this work, we study Fourier multipliers on noncommutative spaces. In particluar, we show a simple proof of $L^p$-$L^q$ estimate of Fourier multipliers on general noncommutative spaces associated with semi-finite von Neumann algebras. This includes the case of Fourier multipliers on general locally compact unimodular groups.

This is an introduction to the algebras $A\subset B(H)$ that the bounded linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mostly interested in the von Neumann algebras, which are stable under taking adjoints, $T\to T^*$, and weakly closed. When the algebra has a trace $tr:A\to\mathbb C$, we can think of it as being of the form $A=L^\infty(X)$, with $X$ being a quantum measured space, and of particular inte...

The purpose of the paper is to establish weighted maximal $L_p$-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the $L_p$ constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singul...

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $\tau$. Let $E(\mathcal{M},\tau) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},\tau)$. We obtai...

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall x\in E(\M)\quad \|T(x...

In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an operator $a\in \mathcal M$ is said to be {irreducible in} $\mathcal M$ if $W^*(a)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(a)'\cap \mathcal M={\mathbb C} \cdot I$. In this paper, let $\Phi(\cdot)$ be a $\Vert\cdot\Vert$-dominating, ...

This paper is mainly devoted to the following question:\ Let $M,N$ be von~Neumann algebras with $M\subset N$, if there is a completely bounded projection $P\colon \ N\to M$, is there automatically a contractive projection $\widetilde P\colon \ N\to M$? We give an affirmative answer with the only restriction that $M$ is assumed semi-finite.